Generalized Inverses

Generalized Inverses

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1. The Inverse of a Nonsingular Matrix It is well known that every nonsingular matrix A has a unique inverse, ?1 denoted by A , such that ?1 ?1 AA = A A =I, (1) where I is the identity matrix. Of the numerous properties of the inverse matrix, we mention a few. Thus, ?1 ?1 (A ) = A, T ?1 ?1 T (A ) =(A ) , ? ?1 ?1 ? (A ) =(A ) , ?1 ?1 ?1 (AB) = B A , T ? where A and A , respectively, denote the transpose and conjugate tra- pose of A. It will be recalled that a real or complex number ? is called an eigenvalue of a square matrix A, and a nonzero vector x is called an eigenvector of A corresponding to ?, if Ax = ?x. ?1 Another property of the inverse A is that its eigenvalues are the recip- cals of those of A. 2. Generalized Inverses of Matrices A matrix has an inverse only if it is square, and even then only if it is nonsingular or, in other words, if its columns (or rows) are linearly in- pendent. In recent years needs have been felt in numerous areas of applied mathematics for some kind of partial inverse of a matrix that is singular or even rectangular.Section 6. Applications to system theory. Balakrishnan [40], Barnett [51], Ho and Kalman [422], Kalman [458], [459], [460], [461], Kalman, Ho and Narendra [462], Kishi [476], Kuo and Mazda [490], Minamide and Nakamura [556], [557], ... Applications to singular linear differential equations and boundary value problems.

Title:Generalized Inverses
Author: Adi Ben-Israel, Thomas N.E. Greville
Publisher:Springer Science & Business Media - 2003-06-16

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